Education ∪ Math ∪ Technology

Year: 2013 (page 11 of 15)

Become a mathematician

I posted this on Twitter a few weeks ago:

In response, someone asked a very sensible question; how do I do this?

 

Here are some ideas.

  • Find an area of mathematics with which you are not currently very familiar, and start exploring it. This might look like a combination of reading what other people have written about this topic, and experimenting with it yourself (mix and match according to your preference). One excellent starting place here I recommend is Paul Lockhart’s Measurement book. Another area of mathematics (or that uses mathematics) that I recommand all math educators learn at least a little bit about is programming.

    Alternative: take an area of interest and see if you can develop your own mathematical framework around it. After all, no mathematical framework existed until someone started exploring patterns and the relationships between objects and then creating a language to describe these relationships.
     

  • Solve the challenging problems from whatever textbook you are using. Don’t rely on the answer key! Ask for help from colleagues if necessary. Getting stuck will help you develop empathy for your students when they get stuck. Try and reflect on how you got "unstuck" and help students learn this process. Another source of challenging problems: past mathematics contests.
     
  • Explicitly look for mathematics. Keep a journal of the mathematical ideas you find. Share what you find with others. Incorporate whatever you can into your classroom. See if you can apply mathematics you know to solving some of the problems you encounter in your daily life.

Social media

Tweet distribution
(Image source: Twtrland)

 

The image above is a pie chart of the distribution of the types of tweets I have. Notice that the bulk of my tweets are replies. If you aren’t having discussions via the media, or not very many discussions, then it’s not really social media.

Preparing Students for an Uncertain Technological Future

I wrote this article for our parents in our school’s monthly magazine a few months ago, and realized I had not yet shared it here.


 


(Image credit: Will Lion)

No one knows exactly what form the technology of the future will take, although there are those attempting to make predictions. Ray Kurzweil, a futurist, has this to say on the accelerating rate of change of technology:

Now back to the future: it’s widely misunderstood. Our forebears expected the future to be pretty much like their present, which had been pretty much like their past. Although exponential trends did exist a thousand years ago, they were at that very early stage where an exponential trend is so flat that it looks like no trend at all. So their lack of expectations was largely fulfilled. Today, in accordance with the common wisdom, everyone expects continuous technological progress and the social repercussions that follow. But the future will be far more surprising than most observers realize: few have truly internalized the implications of the fact that the rate of change itself is accelerating.

We do know that new technologies impact our society in profound ways. Of this issue, Neil Postman, a media theorist, said this of technology change:

Technological change is not additive; it is ecological. I can explain this best by an analogy. What happens if we place a drop of red dye into a beaker of clear water? Do we have clear water plus a spot of red dye? Obviously not. We have a new coloration to every molecule of water. That is what I mean by ecological change. A new medium does not add something; it changes everything. In the year 1500, after the printing press was invented, you did not have old Europe plus the printing press. You had a different Europe. After television, America was not America plus television. Television gave a new coloration to every political campaign, to every home, to every school, to every church, to every industry, and so on.

The introduction of the car influenced both how and where we live, and the environments we live in. The television radically transformed how information and culture are transmitted in our society. Airplanes changed where we travel, and how far away from our extended families we are willing to live. The Internet is in the midst of transforming every aspect of our society, from how we shop, to how we learn new things, to how we work. These disruptive technologies could not have been predicted 120 years ago, and we are still feeling the ripples of the changes these technologies have wrought on our society.

If we know that an unknown change is going to greatly influence our students’ lives, how do we prepare them for it?

The first key to preparing for an uncertain future is adaptability. One has to be willing to explore new technologies and see how those technologies influence us. This is one of the strengths of our ‘bring your own computer’ program. Students see that we accept a wide variety of different computers, and can often be seen exploring and sharing each others’ technology. In this way, they learn that computers are not about one model, or one brand, but instead about the functionality of the computer.

The second key to preparing is critical thinking. This is part of the purpose of the IB education our students experience. They learn to question everything. They learn not to accept ‘facts’ at face value, and to recognize that people use facts to represent their perspective on an issue. The same thing is true of technology change. Most people see technology as this magical thing that influences and improves our lives; our students will hopefully see technology as a thing to be examined and critiqued. The critiques students learn how to do in their IB History class will help them critique their own use of technology. They will hopefully be able to ask two important questions of each new technology they encounter: what abilities does this technology give us, and what does it take away?

A third key to preparing for the future is balance. We are not a technology school, we are a school that uses technology. Our students learn a wide variety of different tools, but they also learn, through our Outdoor Experiential Education and our Creativity, Action, Service programs, about the balance necessary to have a successful life. Our hope is that students will be able to apply this understanding of balance to all aspects of their lives, including their use of technology.

Our students face an uncertain future as our society adapts to as yet unknown technologies. They may see the introduction of flying cars, microscopic computers, artificial intelligence, and other technologies that we cannot even imagine. They will likely experience more radical changes in our society than all of the most disruptive technologies from the past century have managed to do. Our hope is that the education they experience at Stratford Hall helps prepare them for these changes.

Assessment for learning

The director of our senior school, Brad Smith, is doing a workshop today in our staff meeting on assessment for learning. He’s found the following quotes, which he wants to use in his presentation. I’m helping him edit his presentation (since I have time, and he does not), and I’m hoping to find some photos or other prompts which describe these statements to include in his presentation.

 

Many argue that “Formative Assessment” is a misleading term and is open to a variety of interpretations. “Assessment for learning” may be preferable and requires 5 elements to be in place:
 

  1. The provision of effective feedback to students
  2. The active involvement of students in their own learning
  3. The adjustment of teaching to take into account the results of assessment
  4. The recognition of the profound influence assessment has on the motivation and self-esteem of students, both of which are crucial influences on learning
  5. The need for students to be able to assess themselves and understand how to improve

Dylan Wiliam

 

When a teacher teaches, no matter how well he or she might design a lesson, what a child learns is unpredictable. Children do not always learn what we teach. That is why the most important assessment does not happen at the end of learning – it happens during the learning, when there is still time to do something with the information.

Dylan Wiliam, 2011

 

The worst scenario is one in which some pupils
who get low marks this time
also got low marks last time
and come to expect to get low marks next time.

This cycle of repeated failure
becomes part of a shared belief
between such students and their teacher.

Black and Wiliam, 1998

 

The more you teach without finding out who understands the concepts and who doesn’t, the greater the likelihood that only already-proficient students will succeed.

Grant Wiggins, 2006

 

The initiate-respond-evaluate cycle:

I’ll ask the question,
a few of you will answer
for the entire class,
and we’ll all pretend
this is the same thing as learning.

Fisher and Frey, 2007

 

If students left the classroom before teachers have made adjustments to their teaching on the basis of what they have learned about the students’ achievement, then they are already playing catch-up. If teachers do not make adjustments before students come back the next day, it is probably too late.

Dylan Wiliam, 2007

 

For teachers, getting annual test scores several months after taking the test and in most cases long after the students have departed for the summer sends a message: “Here’s the data that would have helped you improve your teaching based on the needs of these students if you would have had it in time, but since it’s late and there’s nothing you can do about it, we’ll just release it to the newspapers so they can editorialize again about how bad our schools are."

Doug Reeves, 1998

 

If you know of any photos or visual prompts/diagrams which you think may be useful, please let me know. His workshop is this afternoon, but I’m sure the resources will be useful another time as well. If I find resources which are useful, I’ll share them here.

Online conference for mathematics education

The NCTM recently sent out survey asking their members how they feel about the timing of their major conference, normally held in April, asking them to consider alternate dates. I sent in my responses, and in the comment section proposed that another reason they may be seeing decline in their major conference (while seeing a simultaneous increase in their regional conferences) is that the money just isn’t in the US system anymore to support as many people attending an expensive conference in another city.

I had another proposal as well: what if the NCTM helped organize an online conference, much along the same lines as the very successful K-12 Online Conference.

I see the structure of the conference being much different than how a typical conference is structured. There could be pre-taped video presentations and keynotes as is typical but there could also be online workshops (in which participants are expected to actually participate and do something), sharing sessions, building sessions (wherein participants collaborate to build something for everyone’s classes), analysis of research sessions, mathematical problem solving sessions, and free-form discussions related to mathematics education (which would be excellent for that all so important opportunities for networking that conferences offer).

While the experience would not be the same, the cost would be so much less that many more people could potentially participate. With the NCTM as a supporting partner for such a conference, I suspect that many thousands of people would get involve. In fact, people from all over the world could attend the conference virtually!

Any thoughts?

 

Update: It occurred to me that a dedicated group could run this conference without the support of a major organization like the NCTM. Anyone interested in exploring this option with me?

Where can I get started?

If you are just getting started with technology integration in your mathematics class, then a sensible question is, where do I get started?

Here are four suggestions.

  1. Introduce a graphing and geometry program into your classroom, like Geogebra or Geometer’s Sketchpad. These programs have the flexibility to allow you graph functions, explore geometry, and much more, all with the same program. Geogebra is a free install but your school may already have a license for Geometer’s Sketchpad. Both programs come with significant user communities and additional support resources.
     
  2. Learn about student blogging. When I started it eight years ago with my students, I had one student each day summarize what happened in class, and I would rotate through students in the class. Other students were asked to comment on each other’s posts. This allowed my students to have an ongoing summary of the class (which they really appreciated for studying for assessments, and when they were absent from class), synthesize what they were learning in class, and when looking at each other’s posts they had another opportunity to see someone else’s perspective on the mathematics they were learning. Each year I used student blogging in my class the explanations students created became better. I left a school one year, and halfway through the following year, one of my former students tracked me down and asked me to put the blog back up so they could use it to study for their exams at the end of their two year course.
     
  3. Use video recording in your classroom, either by recording experiments, having students create explanations, or act as hooks to problem solving tasks. With today’s video editors, learning how to edit video is relatively easy, and most video format compatibility problems are far less of an issue than they used to be.
     
  4. Learn how to program a computer, with your students. First, you will learn a useful skill, and so will your students. Second, your students (and you!) will learn more about decomposing problems, debugging problems, and some potentially highly complex mathematical ideas. There are some excellent resources out there which make learning programming much easier than it used to be so do not feel intitimidated by this task. This is an excellent opportunity to model learning for your students.
     

What other technologies would you recommend math teachers learn how to use early in their technology integration journey?

Everyone should learn to code

Code.org has released a video of some big names in the programming world talking about their first experiences in computer programming, and why they think everyone should learn to code.

The reasons they give are that programming teaches you to think (via a quote from Steve Jobs), it helps you learn how to decompose problems into smaller steps, that learning how to program gives you power, and that it is a fundamental literacy in an age of ubiquitous computing.

I don’t see the "there will be lots of jobs available for you" reason that this video presents as a particularly good reason to learn how to program. What if the jobs that were required were jobs that did not give you power, or did not make you think? If everyone did know how to program, do you think that those offices would still look as cool?

Compare this first video to this video of Seymour Papert talking about computers and learning.

Notice that this video also presents the same reasons as in the first video (aside from future potential for jobs) but it gives agency for learning and presenting these reasons to children. I think that the second video is a more powerful as a result.

The video of Seymour Papert is from about 30 years ago and has a little over 9000 views on YouTube. The Code.org video is from a few days ago and already has more than 3 million views. It would be amazing if we could make this idea of Seymour’s – that computers give students agency in learning – a reality, and if we could give it as much coverage as the first video.

 

* If you are reading this in your email, you can view the videos online here.

Math in music

I’m always on the look-out for ways of finding connections between mathematics and other areas of knowledge. Music is one of the areas of knowledge that I know has some similarities with mathematics, and so I’ve been brainstorming ways one could incorporate music into a mathematics classroom. Here are a few examples.

  1. A musical scale is an example of a sequence (of notes) and could be used to show the idea the order of objects, related to the order of numbers. As each note in an ascending scale is played in sequence, students should be able to hear that the notes have a order, and then you can relate this order to the order we associate with the counting numbers.

     

  2. Introducing students to patterns can also be done nicely with music, either with notes, or with percussion instruments. Here are two sample patterns. One simple activity to do with students here is to have them produce their own different types of patterns.

     

  3. You can also use music to develop some conceptual understanding of skip counting. Often children are taught to count by 2s and 3s but do not necessarily understand what this means. Obviously one should use manipulatives and other techniques to develop this understanding, but here’s an example of how skip counting sounds in music. This example could also be used as an introduction to simple linear functions as well at a later grade.

     

  4. You could introduce students to fractions by comparing relative sizes of different notes. In the example below, the music starts off with 16th notes, followed by 8th notes, quarter notes, half notes, and finally a whole note. Can you hear how obvious the difference is between the notes?

     

  5. Music notes themselves are sound waves, which if you have an oscilloscope, you can visualize directly as you listen to a note. A pure note has a relatively simple associated wave, but notes as played on a music instrument are almost always composed of multiple harmonics (or waves of different frequences added together). This is an example of a capture from a digital oscilloscope. What do you think the seemingly random waves that appear between the notes are from?

    You can also visualize the volume of the notes (by opening up an audio recording of some music being played in a program like Audacity, for example), and notice an interesting drop-off that occurs. If you measure this drop-off closely, it should match an exponential decay function.

    Notice also what the volume of the notes over time looks like when we zoom in on one of them.

    Interesting periodic function

     

  6. Imagine you played one note on the piano at one constant speed, and another note at a different constant speed. After how many notes would you play both notes at the same time? This is an application of the lowest common multiple (provided you express the number of notes played per unit time in lowest terms). Below is a video where one note is being played at a rate of 120 times per minute, and in a different recording, the same note is being played at a rate of 150 times per minute. Do you notice something interesting when both recordings are played simultaneously?

     

     

  7. Another area where mathematics comes into play is in the ratio of the wavelengths of different notes. Karen Cheng does an excellent job of explaining how this relates to why we appreciate some music more than other music.

 

Hopefully these short examples give you some examples of how mathematics and music are related. In another post, I intend to look at musical instruments, and how mathematics can be used to construct them.

 

* Musical scores created with Noteflight. This program has a free demo one can use without signing in, but if you want to save your work, you will need to sign up for a free account.

** If you are viewing this post in your email, none of the videos will be visible, so I recommend reading it online here.

The Number Hunter

Stephanie Glen shared with me this interesting project she is working on. The objective of the project is to produce a series of videos to get students excited about mathematics in much the same way that Bill Nye excites students about science. Here’s Stephanie talking about the project in her own words.

 

Right now Stephanie needs some money to produce a 5 minute promo to show to potential financial backers of the show. Help her out by donating to her Kickstarter page.